Semi-balanced Partitions of Two Sets of Points and Embeddings of Rooted Forests

Let m be a positive integer and let R1, R2 and B be three disjoint sets of points in the plane such that no three points of R1 ∪ R2 ∪ B lie on the same line and |B| = (m−1)|R1|+m|R2|. Put g = |R1∪R2|. Then there exists a subdivision X1∪X2∪· · ·∪Xg of the plane into g disjoint convex polygons such that (i) |Xi ∩ (R1 ∪ R2)| = 1 for all 1 ≤ i ≤ g; and (ii) |Xi ∩B| = m−1 if |Xi ∩R1| = 1, and |Xi ∩B| = m if |Xi ∩R2| = 1. This partition is called a semi-balanced partition, and our proof gives an O(n4) time algorithm for finding the above semi-balance partition, where n = |R1| + |R2| + |B|. We next apply the above result to the following theorem: Let T1, . . . , Tg be g disjoint rooted trees such that |Ti| ∈ {m, m + 1} and vi is the root of Ti for all 1 ≤ i ≤ g. Let P be a set of |T1|+ · · ·+ |Tg| points in the plane in general position that contains g specified points p1, . . . , pg. Then the rooted forest T1 ∪ · · · ∪ Tg can be straight-line embedded onto P so that each vi corresponds to pi for every 1 ≤ i ≤ g.